The bosonic and fermionic propagators under an external electromagnetic field in a Euclidean manifold

In this paper, we will calculate the bosonic as well as fermionic propagators under classical homogeneous and constant magnetic and electric fields in a Euclidean space. For this, we will reassess the Ritus' method for calculating the Feynman propagator.

Clifford Geometric Algebra

The monograph is devoted to the fundamental aspects of geometric algebra and closely related issues. The category of Clifford algebras is considered as the conjugate category of vector spaces with a quadratic form. Possible constructions in this category and internal algebraic operations of an algebra with a geometric interpretation are studied. An application to the differential geometry of a Euclidean manifold based on a shape tensor is included. We consider products, coproducts and tensor products in the category of associative algebras with application to the decomposition of Clifford algebras into simple components. Spinors are introduced. Methods of matrix representation of the Clifford algebra are studied. It may be of interest to students, postgraduates and specialists in the field of mathematics, physics and cybernetics.

Weyl Law on Asymptotically Euclidean Manifolds

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there is a refined Weyl asymptotics with three terms. The proof of the theorem uses a careful analysis of the flow behaviour in the corner component of the boundary of the double compactification of the cotangent bundle. Finally, we illustrate the results by analysing the operator $$Q=(1+|x|^2)(1-\varDelta )$$ Q = ( 1 + | x | 2 ) ( 1 - Δ ) on $$\mathbb {R}^d$$ R d .

Spectral action in matrix form

Quantization of the noncommutative geometric spectral action has so far been performed on the final component form of the action where all traces over the Dirac matrices and symmetry algebra are carried out. In this work, in order to preserve the noncommutative geometric structure of the formalism, we derive the quantization rules for propagators and vertices in matrix form. We show that the results in the case of a product of a four-dimensional Euclidean manifold by a finite space, could be cast in the form of that of a Yang–Mills theory. We illustrate the procedure for the toy electroweak model.

The Generalized Thermodynamical Beltrami – Michell Tensorial Equations for the general Thermodynamical State of the Hooke Body: معادلات بيلترامي- ميشيل التنسورية الترموديناميكية المعممة لأجل الحالة الترموديناميكية العامة لجسم هوك

This paper concerns the mathematical linear model of the elastic, homogeneous and isotropic body, with no considerable structure and with infinitesimal elastic strains, subjected to Thermal effects, in the frame of coupled thermoelectrodynamics; discussed firstly by Hooke (in the isothermal case), and shortly called (H). In this paper, firstly we introduce the invariable tensorial traditional and Lame descriptions of the coupled dynamic, thermoelastic, homogeneous and isotropic Hooke body, which initial configuration forms a simply-connected region in the three dimensional euclidean manifold. The news of this paper consists in deriving the invariable tensorial, generalized Beltrami – Michell stress-temperature equations for the (H) thermoelastic body (in the more general case than the thermal stress state), which initial configuration forms a simply-connected region in the three dimensional euclidean manifold. Finally, we end the paper by suggesting the problem for discussing, in addition to another open problem.

On Homeomorphism Between Euclidean Subspace and Conformally Euclidean Manifold

The article presents the proof of the homeomorphism between Euclidean subspace E6of the classical three-body system and 6D Riemannian manifold M, which allows reducing the dynamical problem to the system of the 6th-order

Topological Assessment of Unidentified Moving Objects

Starting from unidentified objects moving inside a two-dimensional Euclidean manifold, we propose a simple method to detect the topological changes that occur during their reciprocal interactions and shape morphing.  This method, which allows the detection of topological holes development and disappearance, makes it possible to solve the uncertainty due to disconnectedness, lack of information and absence of objects’ sharp boundaries, i.e., the three troubling issues which prevent scientists to select the required proper sets/subsets during their experimental assessment of natural and artificial dynamical phenomena, such as fire propagation, wireless sensor networks, migration flows, neural networks’ and cosmic bodies’ analysis.   

A STRAINED SPACE-TIME TO EXPLAIN THE LARGE SCALE PROPERTIES OF THE UNIVERSE

Space-time can be treated as a four-dimensional material continuum. The corresponding generally curved manifold can be thought of as having been obtained, by continuous deformation, from a flat four-dimensional Euclidean manifold. In a three-dimensional ordinary situation such a deformation process would lead to strain in the manifold. Strain in turn may be read as half the difference between the actual metric tensor and the Euclidean metric tensor of the initial unstrained manifold. On the other side we know that an ordinary material would react to the attempt to introduce strain giving rise to internal stresses and one would have correspondingly a deformation energy term. Assuming the conditions of linear elasticity hold, the deformation energy is easily written in terms of the strain tensor. The Einstein-Hilbert action is generalized to include the new deformation energy term. The new action for space-time has been applied to a Friedmann-Lemaître-Robertson-Walker universe filled with dust and radiation. The accelerated expansion is recovered, then the theory has been put through four cosmological tests: primordial isotopic abundances from Big Bang Nucleosynthesis; Acoustic Scale of the CMB; Large Scale Structure formation; luminosity/redshift relation for type Ia supernovae. The result is satisfying and has allowed to evaluate the parameters of the theory.